Finite generation of iterated wreath products in product action

Abstract

Let S be a sequence of finite perfect transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated wreath product in product action of the groups in S is topologically finitely generated, provided that the actions of the groups in S are not regular. We prove that our bound has the right asymptotic behaviour. We also deduce that other infinitely iterated mixed wreath products of groups in S are finitely generated. Finally we apply our methods to find explicitly two generators of infinitely iterated wreath products in product action of special sequences S. Notation 1. All the actions will be right actions and e will stand for the identity element of a group. We will write n = {1, . . . , n}.Notation 2. Let d be an integer. In all of this paper we will denote by S a sequence of finite transitive permutation groups {S k ≤ Sym(m k )} k∈N such that each S k is perfect and at most d-generated as an abstract group.

Hereditarily just infinite profinite groups with complete Hausdorff dimension spectrum

“…By [Re,Theorem 6.2] and [Va15], every infinitely iterated wreath product w.r.t. product actions W pa (S), based on a sequence S of finite non-abelian simple permutation groups, is a finitely generated hereditarily just infinite profinite group that is not virtually pro-p for any prime p. In [Wi,Va15,Va16,KV] some embedding, generation and presentation properties of such groups have been established, but many of their features are not yet fully understood.…”

On exponentiation, p-automata and HNN extensions of free abelian groups

“…Its basic properties as a permutation group were obtained in [8]. Exponentiation was applied to construct and study new strongly regular graphs ( [15,16]), to study automorphism group of the n-dimensional cube ( [5,6]), to construct new őnite Gelfand pairs ( [2]) and to construct new őnitely generated proőnite groups ( [17]). We observe that exponentiation can be applied to construct groups deőned by őnite automata.…”

Topological finite generation of compact open subgroups of universal groups

“…The question of finite generation of various infinite iterated wreath products was considered by several authors. See [Bha94], [Qui06], [Qui04], [Bon10], [Van15]. All these works consider wreath products of transitive group actions.…”